Chapter 5 – Integrals 5.1 Areas and Distances Dr. Erickson

Consider an object moving at a constant rate of 3 ft/sec. Since rate . time = distance: If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line. time velocity After 4 seconds, the object has gone 12 feet. 5.1 Areas and Distances Dr. Erickson

Velocity is not Constant If the velocity is not constant, we might guess that the distance traveled is still equal to the area under the curve. (The units work out.) Example: We could estimate the area under the curve by drawing rectangles touching at their left corners. This is called the Left-hand Rectangular Approximation Method (L). Approximate area: 5.1 Areas and Distances Dr. Erickson

Right-hand Rectangular Approximation Method (R). Approximate area: 5.1 Areas and Distances Dr. Erickson

Definition #2 – Using Right End Points The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: 5.1 Areas and Distances Dr. Erickson

Definition Using Left Endpoints We get the same values if we use left endpoints. 5.1 Areas and Distances Dr. Erickson

Midpoint Rectangular Approximation Method (M) Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method. In this example there are four subintervals. As the number of subintervals increases, so does the accuracy. Approximate area: 5.1 Areas and Distances Dr. Erickson

width of subinterval With 8 subintervals: Approximate area: The exact answer for this problem is . width of subinterval 5.1 Areas and Distances Dr. Erickson

Definition Using Midpoints We can consider the midpoint to be sample points ( ) so we have the formula: 5.1 Areas and Distances Dr. Erickson

Sigma Notation We often use sigma notation to write the sums with many terms more compactly. For instance, 5.1 Areas and Distances Dr. Erickson

Example 1 Estimate the area under the graph from x=0 to x=4 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or overestimate? Repeat part (a) using left endpoints. 5.1 Areas and Distances Dr. Erickson

Example 2 Graph the function. Estimate the area under the graph from x=-2 to x=2 using four approximating rectangles and taking sample points to be (i) right endpoints and (ii) midpoints. In each case, sketch the graph and the rectangles. Improve your estimates in part (b) using 8 rectangles. 5.1 Areas and Distances Dr. Erickson

Example 3 Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed and values of the rate at two-hour time intervals are shown in the table. Find the lower and upper estimates for the total amount of oil that leaked out. t(h) 2 4 6 8 10 r(t) (L/h) 8.7 7.6 6.8 6.2 5.7 5.3 5.1 Areas and Distances Dr. Erickson

Example 4 Use the definition (#2 according to your book) to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. a) b) 5.1 Areas and Distances Dr. Erickson

Example 5 Determine a region whose area is equal to the given limit. Do not evaluate the limit. 5.1 Areas and Distances Dr. Erickson